solve for X. `(4)/(x+1)le3le (6)/(x+1),(xgt0)`

Solve -2 - (x)/(4) le (1+x)/(3), 3-x lt 4 (x-3) .

solve |x-1|le2 .

Solve for x : "tan"^(-1)(1-x)/(1+x)=1/(2)"tan"^(-1)x,xgt0 .

Solve the inequalities given below for real x :- 3(x -1) le 2 (x -3)

{:(f(x) = cos x and H_(1)(x) = min{f(t), 0 le t lt x},),(0 le x le (pi)/(2) = (pi)/(2)-x,(pi)/(2) lt x le pi),(f(x) = cos x and H_(2) (x) = max {f(t), o le t le x},),(0 le x le (pi)/(2) = (pi)/(2) - x","(pi)/(2) lt x le pi),(g(x) = sin x and H_(3)(x) = min{g(t),0 le t le x},),(0 le x le (pi)/(2)=(pi)/(2) - x, (pi)/(2) le x le pi),(g(x) = sin x and H_(4)(x) = max{g(t),0 le t le x},),(0 le x le (pi)/(2) = (pi)/(2) - x, (pi)/(2) lt x le pi):} Which of the following is true for H_(1) (x) ?

Solve the equation tan^(-1)2x+tan^(-1)3x=(pi)/(4),xgt0

Let f(x) = x^(3) - x^(2) + x + 1 and g(x) = {{:(max f(t)",", 0 le t le x,"for",0 le x le 1),(3-x",",1 lt x le 2,,):} Then, g(x) in [0, 2] is a. continuous for x in [0, 2] - {1} b. continuous for x in [0, 2] c. differentiable for all x in [0, 2] d. differentiable for all x in [0, 2] - {1}

Prove that following: underset(0)overset(9)intf(x) dx, where f(x) = {:{(sinx, if 0 le x le (pi)/(2)),(1,ifpi/2le x le 3),(e^(x-3), if 3 le x le 9)

Discuss the applicability of Rolle's Theorem for the function : f(x) {:{((x^2+1), whe 0 le x le 1),(3-x, when 1 < x le 2):}